FLAC3D Theory and Background • Constitutive Models

Isotropic Elastic Model

In this isotropic elastic (or just, elastic, for simplicity) model, strain increment (\(\Delta{\epsilon}_{ij}\)) generate stress increment (\(\Delta{\sigma}_{ij}\)) according to the linear and reversible law of Hooke:

(1)\[\Delta {\sigma}_{ij} = 2G\Delta e_{ij} + {\alpha}_2 \Delta {\epsilon}_{kk} {\delta}_{ij}\]

where the Einstein summation convention applies, \({\delta}_{ij}\) is the Kroenecker delta symbol, \(\Delta e_{ij}\) is the deviatoric strain increment, and \({\alpha}_2\) is a material constant related to the bulk modulus, \(K\), and shear modulus, \(G\), as

(2)\[{\alpha}_2 = K - {{2}\over{3}} G\]

New stress values are then obtained from the relation

(3)\[{\sigma}_{ij}^N = {\sigma}_{ij} + \Delta {\sigma}_{ij}\]

Bulk modulus, K, and shear modulus, G, are related to Young’s modulus, E, and Poisson’s ratio, \(nu\), by the following equations:

(4)\[K = {E \over {3(1-2\nu)}} \;\;\;\;\;\;\;\; G = {E \over {2(1+\nu)}}\]

or

(5)\[E = {{9KG} \over {3K+G}} \;\;\;\;\;\;\;\; \nu = {{3K-2G} \over {2(3K+G)}}\]

Examples

Simulation of Pull-Tests for Fully Bonded Rock Reinforcement (2D)
Simulation of Pull-Tests for Fully Bonded Rock Reinforcement (3D)
Slope Stability for a Cohesive and Frictional Soil
Excavation in a Saturated Soil
Installation of a Triple-Anchored Excavation Wall

Frequently Asked Questions

1. Q: Can this elastic model be used for water, which has a zero shear modulus?

   A: Yes, you can simulate water behavior by setting bulk modulus equal to the water bulk modulus and shear modules to 0 in the elastic model. An example of this can be found in the Hydrodynamic Pressure Acting on a Dam case study.

2. Q: What is the allowable range for Poisson’s ratio?

   A: The theoretical allowable range for Poisson’s ratio is between -1 and 0.5. However, in practice, the range is typically between 0 and 0.5.

elastic Model Properties

Use the following keywords with the zone property (FLAC3D) or block zone property (3DEC) command to set these properties of the elastic (isotropic) model.

isotropic

bulk f

bulk modulus, \(K\)

shear f

shear modulus, \(G\)

or

young f

Young’s modulus, \(E\)

poisson f

Poisson’s ratio, \(\nu\)

Notes

• Only one of the two options is required to define the elasticity: bulk modulus \(K\) and shear modulus \(G\), or, Young’s modulus \(E\) and Poisson’s ratio \(\nu\). When choosing the latter, Young’s modulus \(E\) must be assigned in advance of Poisson’s ratio \(\nu\).